Multi Reservoir Operating Rules

7/30/2019 Multi Reservoir Operating Rules

1/23

1

Journal of Water Resources Planning and Management, Vol. 125, No. 3, pp. 143-153, May/June 1999

Some Derived Operating Rules for Reservoirsin Series or in Parallel

Jay R. Lund, ProfessorDepartment of Civil and Environmental Engineering

University of California, Davis, CA 95616; [email protected]

Joel Guzmanformer undergraduate student, Department of Civil and Environmental Engineering,

University of California, Davis

AbstractThis paper reviews a variety of derived single-purpose operating policies for reservoirs in series

and in parallel for water supply, flood control, hydropower, water quality, and recreation. Such rules areuseful for real-time operations, conducting reservoir simulation studies for real-time, seasonal, and long-term operations, and for understanding the workings of multi-reservoir systems. For reservoirs in series,several additional new policies are derived for special cases of optimal short-term operation for hydropowerproduction and energy storage. For reservoirs in parallel, additional new special-case rules are derived forwater quality, water supply, and hydropower production. New operating policies also are derived forreservoir recreation.

IntroductionDespite the development and growing use of optimization models (Labadie 1997), the vast majority

of reservoir planning and operation studies are based predominantly or exclusively on simulation modeling,and thus require intelligent specification of operating rules. Practical real-time operations also usuallyrequire the specification of reservoir operating rules. These rules determine the release and storagedecisions for each reservoir at each time-step during the simulation and help guide reservoir operators

(Bower et al. 1966; Hufschmidt and Fiering 1966). This paper reviews a variety of common and newderived operating rules for single-purpose reservoirs in series and in parallel. These derived rules can all besupported by conceptual or mathematical deduction from principles of engineering optimization for specialcases. These rules can be contrasted with the many and often highly effective empirically-based rulescommon in practice, such as various pool-based rules and balancing rules (Wurbs 1996; USACE 1977;Nalbantis and Koutsoyiannia 1997). The rules examined here are intended mostly for seasonal and long-term studies. Real-time studies, with an hourly or daily time-step, often have more detailed safety, habitat,and facility limitations not usually important for studies using coarser time-steps or longer operatinghorizons. The rules presented here offer some guidance for real-time operation, but are more generallyapplicable to seasonal and long-term operations planning and modeling studies. Desireable operating rulesfor reservoir systems with mixed purposes might have very different forms from those presented here.Previous reviews of reservoir operating rules include Sheer's fine concise review (1986) and Loucks and

Sigvaldason (1982). Several Corps of Engineers reports (cited below) also develop reservoir operatingrules for specific purposes. The work presented here is drawn from work done for the U.S. Army Corps ofEngineers (Lund and Guzman 1996).

For those developing operating policies for real reservoir systems, the rules presented are part of abag of tricks with some theoretical or practical basis to recommend them. The rules discussed here areorganized by reservoir configuration, in series or in parallel, and by various operating purposes. Operationfor multiple purposes, i.e., most real systems, requires some combination of these rules or use of otheroperating rule forms (Lund and Guzman 1996). The presumption in many of these rules is that a system of


2/23

2

reservoirs can be operated to produce greater benefits than operating the individual reservoirsindependently. While this is often the case (Palmer et al 1982), it is not always the case (Needham 1998).

The paper begins with examination of rules for reservoirs in series for specific single purposes.Various single-purpose rules for reservoirs in parallel are then reviewed. A general storage-effectiveness-based rule for recreation storage is then presented, followed by a short general discussion of specificationof operating rules by stochastic dynamic programming. The conceptual rules are summarized in Tables 1

and 2. A commentary and conclusions end the paper. Derivations of selected rules appear in appendices.Pseudo-code for implementing many of these rules appear in Lund and Guzman (1996).

Rules for Reservoirs in SeriesThe paper begins with examination of operating rules for reservoirs in series, illustrated in Figure

1, for water supply storage, flood control, energy storage, and hydropower production. Most rulespresented are assembled from earlier cited work. This work is extended in some cases. The conceptualrules for reservoirs in series are summarized in Table 1.

Inflow

Outflow

Reservoir A

Reservoir B

Reservoir CInflow

Inflow

Figure 1: Reservoirs in Series

Table 1: Conceptual Rules for Reservoirs in Series*

Season/Period

Purpose Refill Drawdown

Water Supply Fill upper reservoirs first Empty lower reservoirs first

Flood Control Fill upper reservoirs first Empty lower reservoirs first

Energy Storage Fill upper reservoirs first Empty lower reservoirs first

Hydropower Production Maximize storage in reservoirs with greatest energy production

Recreation Equalize marginal recreation improvement of additional storageamong reservoirs

* Exceptions and refinements are discussed in the text.

Water Storage RulesFor reservoirs in series providing water supply, a reasonable objective is to maximize the amount

of water available, which is the same as minimizing spilled water. The resulting rule for single-purposewater supply reservoirs in series is simply to fill the higher reservoirs first, and the lowest last (Sheer,1986).


3/23

3

The likelihood and severity of shortages is reduced by preventing any water from leaving thesystem as uncontrolled and unproductive spills. For reservoirs in series, with intermediate inflows, theprobability of spill from the system is minimized by first filling the uppermost reservoirs and retainingstorage capacity in the lower reservoirs to capture potentially large flows and reduce the likelihood of spillsfrom the system. Spillage from any but the lowest reservoir can then be captured by a lower reservoir.During the draw-down season, where system inflows are less than demands, the system should be drawn

down in order of the downstream reservoirs first, to provide storage to accommodate potentially excessintermediate inflows or an early onset of the refill season.

For reservoirs in series serving water supply as a sole purpose, the above rule seems universal. Anexception might be where higher reservoirs suffer higher rates of water loss from evaporation and seepage(Kelley 1986). In this case, any increased evaporation or seepage from higher reservoirs would have to beweighed against the increased potential for loss due to spill from concentrating storage at lower elevations.

Flood Control RulesFor reservoirs in series with intermediate inflows and storage serving solely for flood control

downstream, it is optimal to regulate floods by filling the upper reservoirs first and emptying the lowerreservoirs first. The objective is to maintain as much control over flows entering the system above acritical flood-prone reach as possible. Flood storage at the reservoir closest upstream from a critical flood

control reach always provides greater flood controllability than for any other reservoir (Marin et al. 1994).These are typically the lowest reservoirs in the series. Thus, for single-purpose flood control storage in aseries of reservoirs, it is best to fill the higher reservoirs first and empty the lower reservoirs first. Anillustration of this rule can be found in a large Brazilian multi-reservoir system (Kelman et al. 1989).

An exception to this rule can be where the outflow capacity of the lower reservoir is restricted.Here, it can be better to fill the lower reservoir first to increase head on the outlet, thereby increasingrelease capacity from the entire system to the downstream channel capacity (USACE 1976). This allowshigher pre-releases to increase total storage available for a coming flood event.

The operation of reservoirs in series for flood control is fairly complementary with water supplyoperations, at least in regard to the preferred location of storage. The maintenance of water supply storagestill preys on the absolute flood control capability of a system, and vice versa.

USACE (1976) presents methods for allocating flood control space in reservoirs in series with

flood control locations both downstream of the system and between reservoirs. In such cases, except forsmall flood events, there are likely to be inherent trade-offs in protection of these sites from differentstorage allocation and operations decisions.

Hydropower RulesHydropower rules for reservoirs in series vary between refill and drawdown seasons or periods.

During a refill period, the problem usually is to maximize the storage of energy at the end of the period.During a drawdown period, the objective is to maximize hydropower production for a given total storageamount. Different rules are employed for each period. A difficult problem is the transition betweenseasons or periods.

Energy Storage RulesThe objective of the energy storage rule for reservoirs in series is to maximize the total energystored at the end of a refill season or period. Here, the refill season is defined as the season when systeminflows exceed those needed to meet water supply or hydropower production demands. The energy storagerule for reservoirs in series is to always fill the upper reservoirs first.

To maximize the energy stored for a future time, water storage is preferred in upstream reservoirs.Water stored at higher elevations has a higher energy content (kilowatt-hours/unit volume of water stored)than water stored at lower elevations. This is particularly true for water stored in reservoirs in series,where water eventually released from upper reservoirs generates hydropower at the lower reservoirs as


4/23

4

well. Any spills from upper reservoirs are available for capture in space available in lower reservoirs.Kelman, et al. (1989) mathematically examine the allocation of energy storage and flood control storagecapacity in complex multi-reservoir systems. Their results will often indicate the compatibility of thedesirable distribution of energy and flood control storages in such two-purpose systems.

Fortunately, energy storage and water supply storage rules for reservoirs in series are quitecompatible for the refill season, at least in terms of where storage is preferred in the system and their

general intent to accumulate the maximum amount of water. However, with the coming of the drawdownseason, hydropower production rules are required.

Hydropower Production RuleWhen it comes time to produce energy, rules to maximize hydropower production may be

employed. Upper reservoirs generate hydropower by releases which consequently also increasedownstream power generation by increasing heads, if stored downstream, or by subsequent turbine releasesdownstream.

The steady-state hydropower production rule attempts to maximize hydropower generation duringa single time step, given a total storage target for the system, primarily during the drawdown season. Thisproblem involves allocating a given total storage to maximize hydropower production. In general, the rulefavors allocation of storage to those reservoirs which create a higher head per unit volume of storage, have

higher generation efficiencies, and have higher releases, since hydropower production is the product ofhead, efficiency, and release. A variant of this rule is the Corps' "storage effectiveness index" presented inAppendix I (USACE 1985; Lund and Guzman 1996). This rule typically applies to a drawdown period orseason.

The maximum amount of power that can be generated in a reservoir system occurs when headlevels in all reservoirs are at their highest. Where the total amount of water stored in the system is limited,perhaps due to other operating purposes or variations in hydrology and demands, then the problem becomesone of allocating a limited amount of storage among the individual reservoirs to maximize hydropowerproduction. This hydropower maximizing water storage allocation depends on reservoir capacities,inflows, efficiencies of energy production, and the total amount of water (or energy) to be stored.

Water often is stored first in smaller reservoirs, where head usually increases more per unitvolume of additional storage than in most large reservoirs. This is illustrated in Figure 2, where the volume

of water needed to increase head by an amount x in the lower reservoir, V2, is much less than that neededto achieve an equivalent increase in head for the upper reservoir, V1.

x

x

V1

V2

V1 > V2

Figure 2: Change in Head With Varying Capacities

Another consideration is release flow rate. All else being equal, hydropower production ismaximized by allocating available stored water to reservoirs with the greatest release rates. Thus, forreservoirs in series with intermediate inflows, it is often desirable to maximize storage in the lowerreservoirs first. Typically, downstream reservoirs receive more direct and indirect inflows than upstreamreservoirs. Storage in downstream reservoirs is therefore often kept at high levels to take advantage ofincreased flows (with some increased chance of energy spills).

Lastly reservoirs with higher generation efficiencies should be maintained at higher levels ofstorage at the expense of reservoirs with lower efficiencies. The combination of reservoir capacity, amount


5/23

5

of total inflows, and power generation efficiencies determines the overall potential of the reservoir toproduce power, assuming all reservoirs have adequate turbine capacity. When reductions in storage arenecessary, they are made from reservoirs with the least ability to produce power. Conversely, if anincrease in storage can be made, it should be in reservoirs with the greatest ability to produce power.

The interaction of these factors is examined mathematically in Appendix II. The result is tocalculate the following ratio for each reservoir i at each simulation time-step:

(1)

Vi = aiei Ijj =1

i

,where

Vi increased power production per unit increase in storage

ai the unit change in hydropower head per unit change in storage (the slope of the

head-storage curve),ei the power generation efficiency of reservoir i, and

Ij the direct inflows and releases into reservoir j, for all reservoirs upstream of

reservoir i.Here reservoir 1 is the uppermost reservoir in the series of reservoirs. For this steady-statehydropower production rule, reservoirs are ordered in terms of their values of Vi, and are filled

from highest to lowest value of Vi until the total water storage target is met.

For drawdown periods, these rules can be employed for hydropower production systemsof reservoirs in parallel, in series, as well as mixed systems. Where the total storage constraint isdesired to be in terms of energy storage, rather than water storage, then the more elaborate linearprogramming approach presented in Appendix II is required.

Many systems of reservoirs in series maintain their lower, smaller reservoirs full forhydropower production. This is the case for the Missouri River System and the Columbia RiverSystem. Often, maintaining high storage levels in the lower reservoirs also aids navigationthrough the lower reaches of the reservoir system, as in the lower Columbia River System.

The hydropower production rule for reservoirs in series can be implemented directly usingthe values of Vi presented above, or by using the more elaborate linear programming formulations

of the problem presented in Appendix II. This rule should work best for individual time-stepsunder nearly steady-state conditions, where small changes in storage are anticipated.

A more dynamic and general version of this simple rule is discussed by Lund (submitted),based on a concept by Dan Sheer (1986).

Rules for Reservoirs in ParallelThe operation of reservoirs in parallel, Figure 3, differs from reservoirs in series in that

downstream reservoirs cannot be used to capture additional water from underestimated flows orbenefit from the transfer of water stored upstream if flows are overestimated. The following

sections present balancing rules for water supply, energy storage, water quality, and flood control.These rules typically apply to the reservoir system's refill season. Computational studies suggestthat these rules tend to work rather well over a wide range of conditions, perhaps because theperformance response surface is flat for such storage allocation decisions (Sand 1984). Severalhydropower production rules also are presented. Conceptual rules for operating parallelreservoirs are summarized in Table 2.


6/23

6

Inflow A Inflow B

Reservoir A Reservoir B

Water Demand

Figure 3: Reservoirs in Parallel

Table 2: Conceptual Rules for Reservoirs in Parallel*

Season/Period

Purpose Refill Drawdown

Water Supply Equalize probability ofseasonal spill among reservoirs

Equalize probability ofemptying among reservoirs

Flood Control Leave more storage space inreservoirs subject to flooding

N.A.

Energy Storage Equalize expected value (EV)of seasonal energy spill among

reservoirs

For last time-step, equalizeexpected value (EV) of refill

season energy spill amongreservoirs

Water Quality Equalize EV of marginalseasonal water quality spillamong reservoirs

For last time-step, equalize EVof refill season water qualityspill among reservoirs

Hydropower Production Maximize storage in reservoirs with greatest energy production

Recreation Equalize marginal recreation improvement of additional storageamong reservoirs

* Exceptions and refinements are discussed in the text.

Water Supply, Energy Storage, and Water Quality Rules

Several types of rules have been developed for water supply, energy storage, and waterquality operations for parallel reservoirs during refill periods. For reservoir systems providingeither water supply or energy production, a reasonable objective is to minimize expectedshortages. The severity of shortages is reduced by avoiding any water leaving the system asuncontrolled and unproductive spills (Sand 1984). These rules prescribe ideal release or storagelevels for reservoirs in parallel to avoid the inefficient condition of having some reservoirs full andspilling, while other reservoirs have unused storage capacity (Bower et al. 1966). These rules arederived in Appendix III.


7/23

7

New York City Rules (NYC rules)

The NYC rules use the probability of spills rather than the direct amounts of physical spillin the minimization of expected shortages. When the probabilities of spilling at the end of therefill season are the same for each reservoir, it follows that physical spill also is minimized (See

Appendix III). Water supply shortfall is consequently minimized as well.The NYC rule was first stated by Clark (1950) for the New York City water supplysystem, "In operating this system an attempt is made to have the storage in each of thewatersheds, at all times, fall on the same percentage year." The draw from each reservoir isadjusted to equalize the probability of refill by the end of the refill season, about June 1 (Clark1956).

There are three requirements for rigorous application of the NYC rule (Sand 1984):1. The system contains reservoirs operating in parallel;2. The system provides for a single demand downstream of all reservoirs;3. Expected shortages are to be avoided or minimized.

Modified forms of the NYC rule also can handle situations where the unit value of water variesbetween reservoirs but is constant in any individual reservoir. In terms of water supply, thequality of water might affect its unit value. For example, higher total suspended solids (TSS)concentrations correspond to greater treatment costs and lower desirability as a water supplysource. For energy production the value of water is based on the maximum head of eachreservoir, so water contained in reservoirs with greater head has proportionally greater value.

Application of the NYC rule depends on predicted inflows. Thus greater accuracy inthese predictions should yield better results (fewer spills). Since releases are recalculated at eachperiod, a high degree of accuracy in predicted inflows is not critical in the early periods of therefill season. Towards the end of the refill season, reliable flow forecasts become more importantas the chances of spill increase (Bower et al. 1966). Therefore, it is important to have enoughhistorical and watershed data for probabilistic streamflow forecasts.

Optimality also depends on the coefficient of variation of mean monthly flows and the

correlation between flows on adjacent streams (Bower et al. 1966; Sand 1984). The NYC rulehas been found to behave optimally or near-optimally for a wide variety of operating conditionsand system configurations (Sand 1984).

The general form of the NYC rule equates the probabilities of spill at the end of the refillseason adjusted by the unit value of water for each reservoir:

(2) hi Pr[CQi Ki Sfi]= for all i wherehi unit value of water in reservoir i,

CQi cumulative inflow to reservoir i from the end of the current period to the end of

the refill season,K

istorage capacity of reservoir i, assumed to be the same in every period,

Sfi end-of-period storage for the current period for reservoir i, and

a constant across all reservoirs in parallel.The values ofhi depend on water quality and energy storage issues as described later.

Historical data are typically used to establish the cumulative inflows, CQi. Individual reservoir

releases for the current period,Ri, are found by taking the initial storage, Soi, plus expected inflow

for the current period,E[Qi], and subtracting the end-of-period storage, Sfi, that satisfies equation

(2):


8/23

8

(3) Ri = Soi + E[Qi] - Sfisuch that the sum of releases equals a current-period total downstream release target ROT,

(4) i=1

nRi = ROT.

This often requires a search over to find the proper total release.

Water Supply

When the unit value of water, hi, is the same among reservoirs providing water supply, hi

is incorporated into the constant and thus drops from the equation:(5) Pr[CQi Ki - Sfi] = for all i

Water Quality

When the quality of water varies between reservoirs, such as varying total dissolved orsuspended solids concentrations the probabilities of spill are adjusted by hi, the marginal value of

water use minus its treatment cost for each reservoir. Thus, if the marginal value of treated wateruse downstream is $500/ac-ft and the cost of treatment for water from Reservoir 1 is $50/ac-ft, h1

= $450. Such a formulation also might be applied to provide a balancing rule that preferentiallystores water for downstream fish flows.

Energy Storage

For energy storage applications, the probabilities of the potential energy of spill areequated. Therefore the probabilities of spill are adjusted by hi in Equation 2, the full head level

of each reservoir, representing the marginal value of energy lost from spill.

The NYC space rules apply to the refill season of systems of parallel reservoirs and

attempt to minimize the expected value of spilled water. The primary difficulties are specificationof inflow probabilities, computational implementation of the rule (now a minor problem), andpotentially the absence of considering future refill season demands on the inflows into the system.

Space RuleThe space rule seeks to leave more space in reservoirs where greater inflows are expected,

or where greater potential energy of inflows are expected in the case of energy storage (Bower etal. 1966). It is a special case of the NYC rule, seeking to minimize the total volume of spills. Thesame conditions for applicability and optimality apply. The NYC rule becomes the space rulewhen the distributional forms of inflows into each parallel reservoir are the same, withdistributions scaled by their expected value, i.e., the distribution fi(CQi/EV(CQi)), where EV() is

the expected value operator, is identical for all reservoirs (Sand 1984). This derivation appears inAppendix III. The advantage of the space rule over the NYC rule is its direct computation. Likethe NYC rule, the spill minimizing objective implies that this rule is applicable to the system's refillseason. Like the NYC rule, the space rule has been found to behave optimally or near-optimallyfor a wide variety of operating conditions and system configurations (Sand 1984; Wu 1988).

Johnson et al. (1991) examined the application of space rules for operating the CentralValley Project in Northern California. In this system, power output is maximized while


9/23

9

maintaining high levels of water supply reliability. This application appeared to offerimprovements over simulated operation of the system.

The particular form of the equal ratio space rule depends on the reservoir purpose beingexamined for the system. Space rules have been developed for water supply storage and energystorage purposes.

Water Supply

For water supply purposes, implementing the space rule consists of setting target storagesin each reservoir so that the ratio of space remaining at the end of the current period to theexpected value of remaining refill season inflow for each reservoir is identical (Johnson et al1991). This is expressed mathematically as,

(6)

Ki SfiEV(CQi)

=Ki

i=1

n

V

EV(CQi)i=1

n

, i,

with

(7) V =i=1

n(Soi + EV(Qi)) - ROT

where V = the total water storage of the system at the end of the current time-step and all otherterms are as defined in the NYC rule. Using the above equation, releases in a parallel system ofreservoirs containing equally valued units of water are determined similarly to implementing theNYC rule,

(8) Sfi = Ki -

i=1

nKi - V

i=1

nEV(CQi)

EV(CQi).

Energy Storage

When preventing energy spills, the space rule equation used for water supply is modifiedby replacing reservoir capacities, available storage, and expected inflows with their potentialenergy counterparts. These consist of maximum potential energy that can be stored or thecapacity of the turbines, available potential energy storage, and potential energy of expectedinflows, respectively (Johnson et al. 1991). The substitution of these elements yields thefollowing equation:

(9)

KEi E

fi

EV(CEi)

=KEi

i=1

n

E

EV(CEi)

i=1

n

, i,

whereKEi the maximum energy content of reservoir i,

Efi the target energy content of reservoir i at the end of the current time step,


10/23

10

CEi the cumulative energy inflow to reservoir i,

E the total target energy content of the reservoir system at the end of the currenttime step, and

EV() the expected value operator.If releases or storages fall outside their permitted upper or lower bounds, the decision variables

can be set to those bounds while the remaining variables are balanced according to the space rule(Stedinger et al. 1983). Johnson et al. (1991) use a quadratic program to implement their energyspace rule for each time step for their California Central Valley Project simulation model.

Space rules are simpler to implement than NYC rules. However, they rest upondistributional assumptions which might not always hold. The importance of these distributionalassumptions can be tested for particular situations using long-term simulation modeling.

Linear Program NYC RuleLike the two previous balancing rule forms, the linear program (LP) NYC rule also relies

on historical data to determine expected inflows. However, rather than producing cumulativeinflow distributions for CIi, the inflow data are entered directly into a linear program. Spill, or the

value of spill, is minimized by considering all individual cumulative inflows from each period tothe end of the refill cycle in past years. Compared to the NYC rules, the primary advantage ofLP-NYC rules is its ability to incorporate other (linear) short-term reservoir operation constraintsinto the rule. Such additional constraints might include minimum or maximum flows downstreamof each reservoir or required diversions below a subset of reservoirs.

The same conditions required for the NYC rule apply to the LP-NYC rule. LP-NYC rulesare slightly more general than NYC rules in that they also can incorporate other linear operatingconstraints in the setting of short-term storage targets for each reservoir and could alsoincorporate other operating purposes in the objective function (Johnson et al. 1991). However,implementation of LP-NYC rules require greater computational effort. For each time-step, thelinear program described in Appendix III would be solved. The values ofhi in the formulation

depend on whether water supply, water quality, or energy storage issues are being considered.Thus, water supply, water quality, and energy storage versions of the LP-NYC rule can bedeveloped.

Flood Control Balancing RuleThe approach taken for flood control in parallel reservoirs is to maintain a balance

between reservoirs in terms of occupied capacities and flood runoff from drainage areas. If areduction in outflows is required, it is made from the reservoir with the least percentageoccupancy or smallest flood runoff. When an increase in releases is possible, it is made from thereservoir with the greatest capacity occupied or where relatively higher flood runoff is occurring.Higher releases from reservoirs receiving greater flood-runoff may thus be counterbalanced by

reducing releases from reservoirs receiving lesser runoff (Ghosh 1986).The intent of the flood control balancing rule is to operate the parallel reservoirs to

balance the amount of flood control storage available, while maximizing undamaging releasesfrom the system. While the principle of balancing flood control storage on parallel reservoirsshould be clear, operation to meet this objective is not exact. If the objective were to minimizethe expected value of damaging spills above the downstream channel capacity, then flood controlrules could be developed analogous to the New York City rules. Unfortunately, the objective offlood control is more likely to be minimization of peak downstream flood flows during the refill


11/23

11

season, where peak inflows to the system can arrive during a very short time. This situation isless rigorously represented by the NYC rule approach.

USACE (1976) suggests the following method for allocating flood control space betweentwo parallel reservoirs (A and B) with a single downstream flood damage location.

1) Route the project design flood and other observed floods with a maximum amount of

runoff occurring above reservoir A and with maximum non-damaging releases from reservoir A.Allow reservoir B to make the remaining releases, up to the maximum non-damaging level. Plotthe space required at reservoir A versus total space required.

2) Perform the same exercise for reservoir B, with the maximum design and observedflood flows entering above reservoir B. Plot the space required at reservoir B versus total spacerequired.

3) The ratio for balancing flood storage between the two reservoirs should lie betweenthese two curves.

While the concept of a flood control balancing rule appears conceptually sound, exactformulations remain unclear.

Water Supply DrawdownFor drawdown of water supply reservoirs in parallel, Wu (1988) suggests a rule which

equalizes the probability of each reservoir being empty at the end of the drawdown season. Forsystems of reservoirs in parallel with side demands, depending on releases from a specificindividual reservoir, such operation is meant to avoid the possibility of having to short a sidedemand when water is available to meet other demands. Wu fashions a drawdown rule alongthese lines similar to the space rule. In simulation tests, he finds a combination of space rule andthis drawdown rule to be simple to implement and providing near-optimal performance relative toother rule forms.

Hydropower Production Rules for Reservoirs in ParallelSteady-State Hydropower Production Rule

For steady-state hydropower production at reservoirs in parallel, the hydropowerproduction rule for small time-steps derived in Appendix II reduces to:

(16) Max P = j=1

mej aj Sj Ij.

where the subscript j refers to an individual parallel reservoir, and variables are defined as forEquation 1. Taking the first derivative, P/ Sj = ej aj Ij = Vj, or the hydropower storage

effectiveness of parallel reservoir j. The resulting rule is: Empty parallel reservoirs sequentially,beginning with those with the smallest Vj. Fill in the reverse order.

Power Production and Energy Drawdown Rules for Parallel Reservoirs

Sheer (1986) provides a rule for drawing down reservoirs with the minimum impact onlong-term hydropower production (e.g., lost potential energy) for reservoirs which will fill beforethey empty. Reservoirs for which withdrawal results in the smallest reduction in potential energyshould be drawn down first. This rule is derived and elaborated by Lund (submitted). Some ofthese results for parallel reservoirs appear below.


12/23

12

Assuming the reservoirs do not empty before they refill, the most efficient drawdown ofwater volume from the system would minimize total reduction in annual hydropower production.The marginal economic value of hydropower release is calculated for each reservoir,

(17) z/ Ti = ei( )P0Hi(Si0) + PrRi H- i(Si0)/ Ti - PfHi(Ki)i .where,ei is the efficiency of power generation at reservoir i,

Ti is the volume of storage to be released from reservoir i in the present time-step,

H-

i(Si0) is the expected flow-weighted hydropower head for reservoir i until refill with present

reservoir storage Si0,

Hi(Ki) is the hydropower head with storage at refill capacity Ki,

P0 is the present price of energy,

Pr is the flow-weighted average price of energy expected until the reservoir refills,

Pfis the expected price of energy when the reservoir is filled, and

i is the marginal proportion of additional storage in the present which would not be spilled

during the refill season for reservoir i (if = 0.9, 10% of any additional storage now is expected tobe spilled).The rule then is to draw down reservoirs with the greatest values of z/ Ti first, and

refill them in the reverse order. (Note that the last two terms are negative and a positive z/ Ti

indicates increased hydropower value with increased release.) This rule is particularly applicablewhere the withdrawals are being made to supply some downstream water supply volumerequirement.

If the current drawdown is intended to supply an energy demand or contract, then theabove rule is modified somewhat. The most efficient drawdown of potential energy from thesystem would minimize total reduction in annual hydropower production. The marginal economicvalue of energy release from each reservoir is estimated, z/ Ei, where Ei = Hi(Si0)eiTi. This

leads to equation 18:

(18) z/ Ei = P0 +PrRi

Hi(Si0) H

-i(Si0)

Ti-

PfHi(Ki)

Hi(Si0) i.

The rule then is to draw down reservoirs with the greatest values of z/ Ei first, and refill them

in the reverse order.Both hydropower production rules should apply well where the reservoirs refill in most

years and do not empty. Under these circumstances, energy spills might be common unlesssufficient turbine flow capacity exists to pass common high refill-season flows. Thus, thecoefficient i can be important. Alternative rules are developed for systems where reservoirs areexpected to empty before they refill (Lund, submitted). Since the value of hydropowerproduction often varies seasonally, these formulations also allow consideration of relative energyprices in different periods.

Storage Allocation for Reservoir RecreationWhere reservoir recreation is the predominant purpose for operations during a time-step,

how should a given total storage be allocated among reservoirs? For individual reservoirs,recreation potential usually varies discontinuously around storage levels corresponding to theelevations of docks, boat-ramps, and beaches. However, within these ranges reservoir recreation


13/23

13

potential is likely to be roughly proportional to reservoir surface area (Ai), perhaps weighted by

some constant representing accessibility or recreational facilities (ri) at each reservoir. Thus, the

systemwide reservoir recreation objective is to maximize total weighted surface area over all nreservoirs:

(19) Max AT = i=1

n

ri Ai(Si)

Subject to:

(20) i=1

nSi = S

Solving this problem using Lagrange multipliers gives the condition that for any two reservoirs iand j,(21) ri Ai(Si)/ Si = rj Aj(Sj)/ Sj,

or, each reservoir's marginal storage contribution to recreation should be equal. This becomes astorage allocation rule for an arbitrary system of reservoirs predominantly used for reservoir

recreation. Since reservoir surface area is usually a concave function of storage, the rule usuallyshould roughly balance storage among all reservoirs, skewed by the weighting factor ri. This type

of storage effectiveness rule is likely to be of greatest use during the drawdown season, whenstorage is decreasing and the distribution of remaining storage becomes an important recreationalissue.

During the drawdown season, operations for hydropower and recreation might be mademore compatible by varying the development of reservoir recreation facilities and access. Thiswould vary ri to make the outcomes of hydropower and recreation rules more closely agree.

Stochastic Dynamic Programming-Based RulesRules for operation of multi-reservoir systems also can be developed by stochastic

dynamic programming (SDP). Here, an explicit characterization of streamflow probabilities isused together with an explicit loss function and definition of system configuration and constraintsto numerically derive optimal reservoir operating policies. This approach has long been exploredand developed (Little 1955; Tejada-Guibert et al. 1995; Kim and Palmer 1997). Variousformulations have been developed to include probabilistic streamflow forecasts and improvementsand approximations to improve computational speed.

SDP approaches to deriving operating rules are rigorous and conceptually flexible.However, they suffer from extreme computational demands for large problems and requireexplicit probabilistic characterization of unimpaired streamflows. Attempts to reducecomputational requirements by increasing the coarseness of storage and flow discretizations leadto results being more approximate (Klemes 1977). In most cases, it also is difficult to have

confidence in a specific explicit probabilistic characterization of a region's inflows and there seemto be a limited variety of approches available for representing streamflow relationships within aSDP format. SDP-based rules continue to show increasing promise, but remain somewhat toprohibitively difficult to apply in practice.

A few tests of SDP versus other operating rules have been performed (Johnson et al 1991;Wu 1988; Karamouz and Houck 1987). These tests show that well-crafted conventionaloperating rules typically perform nearly as well, and sometimes better, than SDP-based rules.


14/23

14

CommentaryRefill versus Drawdown Periods

The rules presented here apply mostly to either periods of refill or drawdown. NYC andSpace rules for reservoirs in parallel and water supply, energy storage, and flood control rules forreservoirs in series typically apply to refill seasons. Hydropower production rules apply mostly to

drawdown periods. Within each type of period, it is relatively easy to determine desirable single-purpose operating policies for the system. It is far more difficult to determine operationally whenthese periods begin and end, and thus how operations should make the transition between theseperiods. Rules for determining which season applies can be based on time of year or streamflowforecasts, and evaluated using historical probabilities, explicit optimization, implicit optimization,and/or historically-based simulation studies. Forecasting is likely to be important here.

Multi-Purpose and Generic Multi-Reservoir RulesMost reservoir systems are multi-purpose. Fortunately, many of the operating rules

presented here show a compatibility between different reservoir purposes, such as flood control,water supply, and energy storage for refill periods on reservoirs in series. In complex cases,

optimization approaches and traditional simulation may be used to calibrate the rule formsidentified above or parameterized simplifications of such multi-reservoir rules. A wide variety ofparameterized forms of these and other multi-reservoir operating rules are available, including theCorps' highly flexible "index-level method" for allocation of total storage among multiplereservoirs (USACE 1977; Nabantis and Koutsoyiannis 1997).

Traditionally, iterative simulation methods are used to calibrate operating rules to performwell for multiple purposes and remain the final analysis method used to refine and test operatingrules (UASCE 1977). Performance-based optimization models also can help sort through theoperations of complex multi-purpose systems. Where implicit stochastic optimization is used forcomputational convenience (Lund and Ferreira 1996), some of the rule forms suggested heremight be useful for disentangling the results. Some of these rule forms also may be useful forother simulation-based reservoir optimization studies, such as genetic algorithms (Oliveira andLoucks 1997).

Reservoir Aggregation in Multi-Reservoir SystemsFor systems consisting of many reservoirs, it is common to aggregate some of the

reservoirs to reduce the computational or data storage demands of large simulation andoptimization models (Saad, et al. 1994). A final use of the rules presented here is to helpaggregate reservoirs and understand how to disaggregate operations of aggregated reservoirs. Inthis regard, reservoirs in series are typically easier to aggregate and disaggregate than reservoirs inparallel. A related use of these rules is to derive improved starting-point solutions for applicationof large scale optimization methods or more refined simulation studies.

Simulation and OptimizationIn almost every case, simulation modeling is the standard by which operating rules are

refined and tested. Simulation models can provide more realistic and detailed representation ofreservoir system operations and much lower computational demands than optimization models forall but the most straight-forward cases. Simulation models also are more common in practice, andtherefore are more likely to be trusted as a standard of comparison.

Comparison of proposed operating rules by simulation modeling provides many benefits.In many cases, simulation results will show that several sets of operating procedures will provide


15/23

15

approximately equivalent performance (Wu 1988). In other cases, simulation results willdemonstrate the tradeoffs of multiple performance objectives with different operating policies. Ina few cases, the ability to demonstrate tradeoffs will aid in negotiations over how the systemshould best be operated. For all these purposes, it is useful to have a wide variety of potentialoperating rule forms available for examination.

ConclusionsDerived operating rules available for reservoirs in series and in parallel have been

reviewed. These rules are summarized in Tables 1 and 2. A wide variety of operating rules areavailable for simulation modeling of reservoirs in series and in parallel. These provide a great dealof flexibility in the specification of system operations under various flow, storage, and demandconditions. Fortunately, for many single-purpose cases, a derived basis exists to help engineersnarrow the search for appropriate operating rules.

A particular set of operating rules can be supported technically in a number of ways.Some rules are based on simple engineering principles for reservoir operations, such as keepingreservoirs full for water supply or empty for flood control. Several rules are derived from more

formal optimization principles, such as the New York City space rules and hydropowerproduction and energy storage rules. However, many rules in practice are based largely onempirical or experimental successes, either from actual operational performance, performance insimulation studies, or optimization results. These experimentally-supported rules are common forlarge multi-purpose projects.

Many opportunities exist for the use of formal optimization methods within reservoirsimulation models. Examples include implementing storage allocation rules for reservoirs in seriesand in parallel, as well as general penalty-minimizing operations and allocating water among useswithin a given time-step. Effective means of employing and hybridizing these rules are likely to berequired for multi-purpose systems. Similarly, rules for shifting operating rule sets according todrawdown or refill conditions are an important area for further work.

Acknowledgments: This work was supported by the US Army Corps of Engineers HydrologicEngineering Center and benefited considerably from discussions with Dan Barcellos, RichardHayes, Morris Israel, Ken Kirby, Dan Sheer, Jery Stedinger, and Mimi Jenkins. Thanks also go totwo reviewers for their helpful comments.

ReferencesBower, B.T., M.M. Hufschmidt, and W.W. Reedy (1966), "Operating Procedures: Their

Role in the Design of Water-Resource Systems by Simulation Analyses," in A. Maass, et al.,Design of Water-Resource Systems, Harvard University Press, Cambridge, MA, pp. 443-458.

Clark, E. J. (1950), New York control curves, Journal of American Water Works

Association, Vol. 42, No. 9, pp. 823-827.Clark, E.J. (1956), "Impounding Reservoirs,"Journal of the American Water WorksAssociation, Vol. 48, No. 4, April, pp. 349-354.

Ghosh, S. N (1986), Flood Control and Drainage Engineering, Oxford & IBH PublishingCo., pp. 158 - 159.

Hufschmidt, M.M. and M.B. Fiering (1966), Simulation Techniques for Design of Water-Resource Systems, Harvard University Press, Cambridge, MA.


16/23

16

Johnson, S.A., J.R. Stedinger, and K. Staschus (1991), "Heuristic Operating Policies forReservoir System Simulation," Water Resources Research, Vol. 27, No. 6, May, pp. 673-685.

Karamouz, M. and M.H. Houck (1987), "Comparison of stochastic and deterministicdynamic programming for reservoir operating rule generation," Water Resources Bulletin, Vol.23, No. 1, pp. 1-9.

Kelley, K.F. (1986),Reservoir Operation During Drought: Case Studies, ResearchDocument No. 25, Hydrologic Engineering Center, Davis, CA.Kelman, J., J.M. Damazio, J.L. Marin, and J.P. daCosta (1989), "The Determination of

Flood Control Volumes in a Multireservoir System", Water Resources Research, Vol. 25, No. 3,March, pp. 337-344.

Kim, Y.-O. and R.N. Palmer (1997), "Value of seasonal flow forecasts in Bayesianstochastic programming,"Journal of Water Resources Planning and Management, ASCE, Vol.123, No. 6, pp. 327-335.

Klemes, V. (1977), Discrete representation of storage for stochastic reservoir operation,"Water Resources Research, Vol. 13, No. 1, pp. 149-158.

Labadie, J. (1997), "Reservoir system optimization models," Water Resources Update,University Council on Water Resources, Number 108, Summer, pp. 83-110.

Little, J.D.C. (1955), "The use of storage in a hydroelectric system," OperationsResearch, Vol. 3, pp. 187-197.

Loucks, D.P. and O.T. Sigvaldason (1982), "Multiple-reservoir operation in NorthAmerica," in The Operation of Multiple Reservoir Systems, Z. Kaczmarek and J. Kindler (eds.),International Institute for Applied Systems Analysis, Laxenburg, Austria.

Lund, J.R. (submitted), "Power Production And Energy Drawdown Rules ForReservoirs,"Journal of Water Resources Planning and Management, ASCE.

Lund, J.R. and I. Ferreira (1996), "Operating Rule Optimization for the Missouri RiverReservoir System,"Journal of Water Resources Planning and Management, ASCE, Vol. 122,No. 4, July.

Lund, J.R. and J. Guzman, (1996), "Developing Seasonal and Long-term Reservoir

System Operation Plans using HEC-PRM", Technical Report RD-40, Hydrologic EngineeringCenter, U.S. Army Corps of Engineers, Davis, CA, June.

Marin, J.L., J.M. Damzio, and F.S. Costa (1994), "Building flood control rule curvesfor multipurpose multireservoir systems using controlability conditions," Water ResourcesResearch, Vol. 30, No. 4, April, pp. 1135-1144.

Nalbantis, I. and D. Koutsoyiannis (1997), "A parametric rule for planning andmanagement of multiple-reservoir systems," Water Resources Research, Vol. 33, No. 9,September, pp. 2165-2178.

Needham, J.T. (1998),Application of Linear Programming for Flood Control Operationson the Iowa and Des Moines Rivers, MS Thesis, University of California, Davis.

Oliveira, R. and D.P. Loucks (1997), "Operating rules for multireservoir systems," Water

Resources Research, Vol. 33, No. 4, April, pp. 839-852.Palmer, R.N., J. Smith, J. Cohon, and C. ReVelle (1982), "Reservoir management in the

Potomac River Basin,"Journal of Water Resources Planning and Management, ASCE, Vol. 108,No. 1, pp. 47-66.

Saad, M., A. Turgeon, P. Bigras, and R. Duquette (1994), "Learning DisaggregationTechnique for the Operation of Long-Term Hydroelectric Power Systems," Water ResourcesResearch, Vol. 30, No. 11, November, pp. 3195-3203.


17/23

17

Sand, G.M. (1984),An Analytical Investigation of Operating Policies for Water-SupplyReservoirs in Parallel, Ph.D. Dissertation, Cornell University, Ithaca, NY.

Sheer, D. (1986), "Dan Sheer's Reservoir Operating Guidelines," as interpreted by J.R.Stedinger, School of Civil and Environmental Engineering, Cornell University, June/July, 1 page.

Stedinger, J. R., B. F. Sule, and D. Pei (1983), Multiple reservoir system screening

models, Water Resources Research, Vol. 19, No. 6, pp. 1383-1393.Tejada-Guibert, J.A., S. Johnson, and J. Stedinger (1995), "The value of hydrologicinformation in stochastic dynamic programming models of multi-reservoir systems," WaterResources Research, Vol. 31, No. 10, pp. 2571-2579.

U.S. Army Corps of Engineers (1976), Flood Control by Reservoirs, HydrologicEngineering Methods for Water Resources Development, Volume 7, Hydrologic EngineeringCenter, Davis, CA, February.

U.S. Army Corps of Engineers (1977),Reservoir System Analysis for Conservation,Hydrologic Engineering Methods for Water Resources Development, Volume 9, HydrologicEngineering Center, Davis, CA, June.

U.S. Army Corps of Engineers (1985),Engineering Manual: Engineering and Design,Hydropower, EM 1110-2-1701, U.S. Army Corps of Engineers, Washington D.C.

Wu, Ray-Shyan (1988),Derivation of Balancing Curves for Multiple ReservoirOperation, M.S. Thesis, Civil and Environmental Engineering, Cornell University, Ithaca, NY.

Wurbs, R.A. (1966),Modeling and Analysis of Reservoir System Operations, Prentice-Hall PTR, Upper Saddle River, NJ.

Appendix I"Storage Effectiveness Index" Rules for Hydropower Production

The "storage effectiveness index" has been developed by the U.S. Army Corps ofEngineers for maximizing firm hydropower production during the drawdown season (USACE,1985). For each reservoir, a "storage effectiveness index" is calculated for each time-step, usingforecast inflows and power demands for the current time-step and remaining time-steps in the

drawdown season. Reservoirs with a low index value are drawn down first.Step 1: Find the firm energy requirement for the current time-step, Ef.

Step 2: Estimate the shortfall of firm hydropower production due to insufficient inflows to thesystem.

Sf = Ef 720

11.81IUiHi Si( )

i=1

n

ei,

where,Sf= energy shortage for the current time-step,

IUi = inflow upstream of reservoir i during the current time-step,

Hi(Si) = hydropower head as a function of reservoir storage for reservoir i,

Si = current reservoir storage for reservoir i,

ei = the hydropower production efficiency of reservoir i, and

the constant is a conversion factor for IUi in cfs, Hi in feet, and Sfin Kwh.

This assumes that all flow can be utilized through the turbines.Step 3: For each reservoir, estimate the drawdown required for that reservoir to individuallyeliminate the shortfall.

Sf =720

11.81(59.5)SiHiei

,


18/23

18

where,1/59.5 = conversion of cfs to acre-ft draft per month,

Si = drawdown, in acre-ft., andHi = an average head corresponding to the drawdown (often found iteratively).

Solving for Si:

Si =11.81(59.5)

720Sf /(Hiei)

.Step 4: For each reservoir, estimate the energy loss in the remainder of the drawdown season due

to a drawdown ofSi during this time-step (month).

ELi

=720

11.81(CI

Ui+ V

pi)H

i(S

i S

i)e

i/59.5 ,

where,

ELi = drawdown season power loss due to drawdown of reservoir i by Si,CIUi = the cumulative natural inflow upstream of reservoir i for the remainder of the refill

season, andVpi = the volume (acre-ft) of upstream storage to be emptied during the remainder of the

drawdown season.Step 5: Calculate the storage effectiveness ratio for each reservoir i:

SERi = ELi /Sf.

Reservoirs with the lowest ratios are to be drawn down first.

Appendix II: Derivation of Hydropower Production Rules forReservoirs in Series

Linear Programming Short Term Storage AllocationThe objective is to find the allocation of a total storage volume which maximizeshydropower production for one period, subject to inflow forecasts for each reservoir, reservoirstorage capacities, and a total storage target. As an equilibrium analysis, changes in reservoirstorage are neglected. This is expressed mathematically below.

(1) Max P = Hi(Si)Qieii =1

n

Subject to:

(2) Q1 = I1,

(3) Qi = Qi1 + Ii , i > 1

(4) Si Ki , i (5) Si

i=1

n

= S

Definition of Variables:P sum total of energy produced by all reservoirs,n number of reservoirs in system,

Hi level of head in reservoir i,


19/23

19

Si storage target for reservoir i,

S Total system storage target,Qi total inflow and release for reservoir i,

Ii direct inflows into reservoir i,

Ki storage capacity of reservoir i,

ei efficiency of turbines in reservoir i,Vi change in overall power production, P with change in storage in reservoir i.

unit weight of waterNote: reservoir i =1 is most upstream reservoir in series.

For short term allocation, the head-storage relationship can often be linearized, orHi(Si) = aiSi.

The following linear program results:

(6) Max P = aiei SiQii =1

n

Subject to constraint Equations (2) - (5)

where ai is a constant relating change in head with change of storage in reservoir i (for small

changes in head). Hi(Si) is commonly non-linear, but may be piece-wise linearized and solvedwith a linear program, since head-storage relationships for most reservoirs are concave.

Derivation of Hydropower Production RuleThe above linear programming formulation can typically be simplified, where the head-

storage relationship can be linearized. Using objective function (6) and substituting in equations(2) and (3) results in the simpler linear program.

(7) Max P = aiei Ijj =1

i

Si

i=1

n

Subject to constraint equations (4) and (5). This problem is solved by finding the slope of the

objective function with respect to storage for each reservoir,

(8)

PSi

aiei Ijj =1

i

= Vi

.Rule: Fill reservoirs in order of highest to lowest Vi until total storage, S, is filled. Empty in the

reverse order.

Linear Programming Short Term Energy Storage AllocationThis problem is slightly modified from that above in that instead of seeking to maximize

hydropower production subject to a given total water storage, it is desired to maximizehydropower production of reservoirs in series subject to a given total energy storage. For this

problem, the objective and constraints in Equations 7 and 4 are re-stated below, and Equation 5modified to an energy storage constraint.

(9) Max P = aiei Ijj =1

i

Si

i=1

n

Subject to:

(10) Si Ki, i


20/23

20

(11)iE

i =1

n

iS( ) = E,

where E is the total energy storage sought and Ei(Si) is the energy content of each reservoir as a

function of it water storage.For small changes in reservoir storage, the function Ei(Si) can probably be linearized into

the form bi Si (where bi is a constant), allowing Equation 11 to be made linear and Equations 9-

11 to be employed as a linear program to allocate storage among reservoirs in series to maximizehydropower production, subject to a total energy storage level. For large changes in storage,another solution method is likely to be required, since Ei(Si) is convex.

Appendix IIIDerivations of Parallel Balancing Rules

These derivations of the New York City and Space rules are adapted from derivationspresented by Sand (1984) and Johnson, et al. (1991). These derivations are further extended toexamine energy storage and water quality applications.

Basic Derivation of the New York City RuleDefinition of Variables:

z value of objective function,hi unit value of water in reservoir i,

n number of reservoirs in the system,Sfi end-of-period storage for the current period for reservoir i,

S0i beginning of current period storage for reservoir i,

CQi the cumulative inflow to reservoir i from the end of the current period to the end

of the refill cycle,Ki storage capacity of reservoir i, assumed to be the same in every period,

V total volume of water in storage at the end of the current period,Ii inflow to reservoir i for the current period,

D demand for current period (release for current period), andfi(CQi) the probability density of CQi.

The objective of the New York City balancing rule is to minimize the expected value EV() of totalcumulative spill from all of the parallel reservoirs at the end of the refill season. This is reflectedin the objective function in Equation 1. In Equation 1, the term hi represents the relative value of

water stored in each reservoir. Variation in the value of water can reflect variation in pumpingcosts, treatment costs, or energy content between the various reservoirs, as discussed later in thederivation. To implement this rule, this optimization problem is solved for each time-step duringthe refill reason.

(1) Min z = EV himin(0,Sfi + CQi Ki )i= 1

n

subject to the following constraints:

(2)S

fi

i =1

n

= V


21/23

21

(3)V= (S

0i+I

i)

i =1

n

D

Constraint Equation 3 indicates that the total water available should equal the sum of availablewater (current storage plus current period inflows) minus downstream water demands for thecurrent period.

Expanding the expected value function in the objective function (Equation 1) yields,

(4) Min z = hi (Sfi + CQi Ki ) fi(CQiKi Sfi

)dCQi

i = 1

n

subject to constraint Equations 2 and 3.The Lagrangian for this problem is:

(5) L = hi

(Sfi

+ CQi

Ki) f

i(CQ

i)

Ki Sfi

dCQi

i= 1

n

+ Sfii = 1

n

V

= hi (Sfi Ki ) fi(CQi)Ki Sfi

dCQi + CQi fi (CQi)Ki Sfi

dCQi

i = 1

n

+ Sfii= 1

n

V

or

(6)L = hii= 1

n

(Sfi Ki) 1 fi(CQi )0

Ki Sfi

dCQi

+ CQi CQi fi (CQi

0

Ki Sfi

)dCQi

+ Sfii =1

n

V

,

where CQi is the expected value of cumulative inflows for reservoir i during the remainder of the

refill season.The first order conditions for solving this problem are:

(7)

LSfi

= 0

=

hi 1 fi(CQi)0

Ki Sfi

dCQi

+ (Sfi Ki )( fi (CQi = Ki Sfi)) (Ki Sfi) fi(CQi = Ki Sfi )

+

or

(8)

hi 1 fi0

Ki Sfi

CQi( )dCQi

=

, or

(9) hi Pr(CQi > Ki - Sfi) = , i, or(10) h

iPr(any spill in reservoir i) = .

This general result indicates that the storage targets for all reservoirs should have the sameprobability of spill, weighted by the value of water for each reservoir, hi. The use of the New

York City Rule for water supply, water quality, and energy storage purposes all follow Equations9 and 10, with different interpretations of hi.

New York City Water Supply Rule


22/23

22

For simple water supply purposes, hihas the same valuefor all i, so Equation (9)

becomes:

(11) Pr(CQi > Ki - Sfi) = , i.

New York City Rule for Water Quality

For simple water supply purposes with important water quality differences (e.g., TDS orperhaps water temperature or TSS) between reservoirs, hivaries between reservoirs and can be

interpreted as the marginal value of water use minus its water treatment cost for reservoir i. Inthis case Equations 9 and 10 remain the same, but with this net water value varying with spills ofdifferent water qualities.

New York City Rule for Energy Storage

Here, the objective is to minimize the expected value of potential energy spilled ratherthan physical water spilled. Here, hi has the interpretation of the energy content of water stored

in reservoir i. Equations 9 and 10 remain the same and applied with this interpretation.

Derivation of Space RulesReturning to Equation 9, the central result of the New York City Rule, the assumption is

made that the distributions fi(CQi) have the same distributional form, except that they are scaled

by the average cumulative flow of the basin, CQi . Where this assumption holds, then the

distributions,

(12)fi CQi / CQi( )= fj CQj / CQj

,for any two reservoirs i and j.

Where this is the case, the ratio (Ki - Sfi)/CQi becomes a standard deviate for all

distributions, having the same probability of exceedence for all reservoirs. If this ratio is set so

that it equals the same ratio at the basin-wide scale,

(13)

Ki

i= 1

n

V

EV(CQi)

i = 1

n

,

then the reservoirs are all balanced in terms of minimizing expected value of spill and maximizingcapture of current inflows.

By replacing water inflows, water storage capacities, and water storage levels with energyinflows, energy storage capacities, and energy storage levels, the space rule can be adapted toenergy storage purposes much as the NYC rule can be adapted to other operating purposes

(Johnson, et al., 1991).

Derivation of Linear Programming-NYC RulesDerivation of the linear programming space rules begins with the Equations (1-3) used in

the derivation of the NYC rules. Additional constraints can also be added to the LP-NYC ruleproblem, so long as the additional constraints are linear.

In this case the expected value operator in Equation 1 is replaced by use of the weighedsummation of spill values that would result from each year of the historical record or the


23/23

probabilities of several representative year-types. Given historical streamflows of equal recordlength >m for each ofn reservoirs, m representative refill seasons can be inferred. This yields thefollowing linear program:

(14) Min z = j=1

mpj

i=1

nhi Lij

Subject to:

(15)S

fi

i =1

n

= V

(16) Lij - Eij = CQij + Ki - Sfi for all i and j,

(17)V= (S

fi+ Q

i)

i =1

n

D

plus any other linear constraints on present-period operationswhere

m number of equally probable refill seasons;

pj probability of hydrologic year-type j;CQij in hydrologic yearj , the expected cumulative inflow to reservoir i from the end of

the current period to the end of the refill cycle;Lij spill from reservoir i under hydrologic yearj; and

Eij empty storage capacity in reservoir i under hydrologic yearj.

Date post:	04-Apr-2018
Category:	Documents
Upload:	pramods8
View:	224 times
Download:	0 times

Multi Reservoir Operating Rules

Documents